Saturday, November 28, 2009

The Causal Diagram of the Black Hole

As I pointed out, though a solution of Einstein's field equations, this diagram does not actually describe a situation we find in reality. The black hole shown in this diagram is accompanied by a white hole, and both have existed since forever, and will continue to exist, unchanging, until eternity. Today, I thus want to discuss the metric for a realistic black hole, a black hole formed from collapse of matter. I will also briefly touch on the evaporation but, as you know if you've been around for a while, the exact way the evaporation proceeds, in particular the final stage, is still under debate.

To obtain the causal diagram of the black hole, recall that Einstein's field equations are local and the black hole solution is a vacuum solution. Yes, that is right. This means that in General Relativity empty space is not necessarily flat. (Flat meaning the curvature tensor vanishes identically. Empty space however has a property called "Ricci-flatness.") If we want to describe collapsing matter, we thus know that outside of that matter the previously found solution, depicted above, still holds. So, what we do is drawing into the diagram the surface of the collapsing matter, and keep the part that is outside that matter. This is shown below.

Now the blue shaded part is the one that no longer correctly describes the black hole that forms from collapse and has to be discarded. This means in particular that the white hole as well as the second asymptotically flat regions are both gone and do not exist in real world situations (addressing a concern that Andrew brought up in the previous post).

What we do then is to attach an interior solution that does not describe vacuum. In some simplified cases this can be done explicitly. For example if the collapsing density is homogeneous (which would be a piece of a FRW-metric), or if it is null dust (described by the Vaidya-metric). Then, one can calculate the interior solution and use a matching condition to join both parts together. For our purposes however, we don't have to bother with the details since we just want to capture the causal structure. For what the causal structure is concerned, the inside solution is rather dull. There is nothing specific going on. The radius just shrinks until it falls below the Schwarzschild radius associated to its total mass. Then the horizon forms, and the matter collapses to a singular point. This is shown in the diagram below.

Note that there is no particular meaning to curves that are exactly horizontal or vertical, we are thus free to deform them, which has been done to make the r=0 curve vertical. This is fine as long as we make sure that the null curves on 45° angles remain the same, and thus spacelike remains spacelike and timelike remains timelike.

As pointed out in the previous post, the use of radial coordinates means that ingoing curves look as if they are reflected at r=0 when they actually go through. The lightray marked v0 in the above figure is the last light ray that just manages to escape the forming horizon. It is in this background, not the static background, that Hawking did his calculation which showed that black holes do emit radiation.

Knowing the black hole, once formed, emits radiation of course brings up the next question: how do we incorporate the evaporation into the diagram? One can add the evaporation of the black hole by using another non-vacuum patch that describes outgoing radiation which leads to a decreasing of the mass. The Schwarzschild-radius of the black hole then gets closer to the singularity until both, the horizon and the singularity, vanish in the endpoint of evaporation. In this process, the event horizon remains lightlike. What changes for the observer at scri minus is the mass associated to the black hole. When the black hole is completely evaporated, we are left with a spacetime filled with very dilute radiation. This spacetime is to good precision flat and described by another piece of Minkowski metric. If you patch the pieces together you get the diagram below.

78 comments:

Yes, by all means continue, Bee! You have a gift for clear expository writing!

Look, Andrew Thomas, see? Bee links to one of your posts! I told you that you're a celebrity, man! We can count the days until the Physics groupies start showing up on your seaside condominium doorstep in Swansea. Be sure to report back to us on the downside of fame, my friend. :-)

Information can "be lost", but no energy. In AWT density gradients (which manifest by gravity field at the case of vacuum) are of dispersive nature and the event horizon of black hole is just a place, where transversal waves of light are getting dispersed into longitudinal gravitational waves in analogous way, like the waves at water surface.

The same situation occurs during fall of meteorite or every energetic photon into Earth's atmosphere: most of transversal waves are getting dispersed here into thermal noise of air molecules and even smaller particles. It means, the lost of information in black holes is not a problem of physics as such - only of few silly theories, which are describing it from biased perspective of transversal waves obstinately.

BTW "expository writings" aren't still "explanatory" ones - I'm pretty sure, no one of readers is able to propose some testable experiments from these naive diagrams, to predict something meaningful the less.

Here's a possible solution to the "black hole/information loss" problem, which you may consider or reject as you see fit.

1. The probability approach to doing physics is often useful, but is not the optimum fundamental approach, which is probably "neo-classical".

2. Perfect reversibility is totally a myth. Everything in nature evolves in an irreversible manner, although for highly controlled and nearly isolated systems there can be approximate reversibility for a limited time.

3. Black holes dominate the Universe and singularities dominate black hole mass distributions. Singularities are the most fundamental and elemental objects in nature. No singularities - no nature.

4. The black hole/information loss "paradox" is based on a poor and/or misguided understanding of how nature actually works.

5. The concept of throwing an encyclopedia into a black hole and retrieving the information is one of those ten impossible things a post-modern physicist must think of before finishing breakfast.

Molecule of semibullvalene in vacuum at a local temp exceeding about 5 kelvins. What is not perfectly reversible about the two-well transition?

Be naughtier given bigger budget. Chiral molecule of norbornenone inserted into a polished superconducting Rabi vacuum resonance cavity with local impedence 1/[(alpha)(8)(pi)(eps_0)(c)], h/[(4)(pi)(e^2)], 2054.11 ohms, then Hund's paradox. What is not reversible about bicyclo[2.2.1]hept-5-ene-2-one racemization?

http://www.mazepath.com/uncleal/norone.png norbornenone stereogram, second structure down

Great article Bee! Thanks so much for the "discarding the blue bit". The resulting diagram might not be so pretty, but at least there's one website with a diagram describing a realistic black hole.

I like the way you describe the inside solution as "rather dull". I'm sure it would be anything but dull if you were actually there!

I'm just trying to get my head round the final diagram. So the singularity remains right up to the point when the black hole finally evaporates? I suppose it does. Is it correct that a small black hole does not have much gravitational pull? Could I have one on my desk as a paperweight?

I suspect you're going to do your next piece on the information loss paradox whether we ask for it or not! But of course we all want it. :)

The last diagram above is the "standard" picture of black hole evaporation. How it "really" looks nobody knows. I will discuss some alternatives in the next post.

As to the gravitational pull, it depends on the distance. A small black hole has less graviational pull than a small black hole at the same distance from the center.

The problem with having one on your desk is that you'd have trouble keeping it there. But besides that, it would have to be extremely small for not to cause you problems. Just to give you some numbers, a black hole with a radius of about 1mm has the mass of about Earthmass. Best,

It’s certainly appreciated all the time and effort you’ve spent in attempting to explain the construction, meaning and utility of causal diagrams. At the same time I must admit to still struggling with this as to be certain I have it all straight. The first thing I’ve noticed, is as you explian the white hole solution here has been eliminated and thus would it be safe to say then precludes the information going from this universe to another or to a different part and or time relevant to this one. The question I would then ask is if this beingb something mandated by reason of logic, resultant and supported by any physical knowledge (experiment or experience) or rather the requirement of particular theoretical choice?

Well, I had been hoping this post was the answer to exactly that question, but let me try it again. The solution with the white hole is a mathematical possible solution to Einstein's field equations. It just does not describe a realistic scenario. To find a realistic scenario, you have to start with initial conditions that one actually finds in Nature. The initial condition for a realistic black hole is some matter distribution that collapses. The solution arising from this does not have a white hole, and it does not have a "paralell universe" (the expression is somewhat misleading in this context though). The reasoning that leads you to this is not mathematical logic, but a physical requirement. Best,

So are you saying a white hole solution is not possible because a black hole is a structure resultant of a condition which forces matter/energy to collapse or rather that it can only be considered if spacetime is is infinite in terms of magnitude and origin? Don't take me wrong for this question is only meant as to clarify.

So is it the time that’s required for it to form being the problem as needing to eternal and infinite or rather time in general. I guess plainly put do you meant to have a white hole solution the black hole would have always had to be as to have had no beginning or end.

So the existence of white holes would have black holes to be structures not formed, yet only things that are and will be forever. Then wouldn’t it equally be valid to argue that a black hole is simply something that never did or never will exist.

I guess what I’m implying here is that all of what takes place at and beyond the event horizon is a matter of theoretical interpretation at limits beyond which the theory extends and not something proven to be mandated by either way of logical or physical consequence. To extend the thought process further, would be to ask if we need to worry about information being lost if the mechanism by which this is taken to happen may not even exist to begin with?

The eternal black hole, as shown in the first diagram, is something that doesn't exist, in the same way that white holes don't exist. We have good evidence however that black holes, as shown in the third diagram, do exist in reality. They are formed by collapsing matter. That such matter under very general conditions inevitably collapses to a singularity and that a horizon forms in that process can be shown rigorously.

The mechanism to throw something into a black hole isn't particularly sophisticated, thus I don't understand your last remark. Best,

I guess what I mean to say is that the information problem could be taken as reason to not have black holes to be considered real in the first place. It just seems to me that this along with the fact that black holes must be made and not simply are and forever, to be consistent and supportive of this hypostasis, rather than contrary to it.

That’s to ask why instead of being driven crazy by having to explain how information can be saved from being lost, to say rather it never is since black holes are in some way prevented from forming as to allow this to happen. That’s to suggest that GR doesn’t begin as being an incorrect explanation only at the boundary formed by the singularity, yet rather at the one formed at the event horizon. It's also to ask why specifically with quantum gravity considerations is GR only considered to have failed at the singularity, rather than at the event horizon, which actually stands as the place where the problem arises.

You like my new avatar? Yeah, that's Buddy Christ from the film "Dogma", in which noted atheist George Carlin plays a Catholic Cardinal (and unveils Buddy C to help reclaim lost faithful), and Alainis Morrisette, who plays God. Lol, I always suspected God was a woman ... in this case one who reacts "not well" to a breakup. ;-)

A bit off topic, maybe, but nevertheless amusing:

The Science Network (television)EPISODE: How to Travel to a Parallel UniverseTuesday, December 1 @ 10:30 PMDr. Kaku is on a mission to design a gateway to a parallel universe – but which type should he visit? MIT cosmologist Alan Guth explains his recipe for creating your own universe in the lab, and physicist Neil Turok explains how a parallel universe is only an atom's length away from us.

And for our next trick: Black Hole paperweights!

Thanks Zephyr for pointing out the difference between "expository" and "explanatory."

First let me say that there have been, and still are, approaches along the lines of reasoning you suggest. I believe we discussed that earlier in context of John Moffat's book in the comments to this post. The reason why the explanation that black holes (as in: their event horizon) don't exist is extremely implausible is (besides the experimental evidence) that it would require a significant modification of General Relativity in regimes where such a modification is not to be expected for any reasons. It would have to be a modification in a background that is flat to arbitrarily good accuracy, like the background you now sit in.

The reason is that the curvature at the black hole horizon can be arbitrarily weak, so are effects of quantum gravity. (Note: Hawking evaporation is not a quantum graviational effect.) This is a very common misunderstanding, so let me spell it out one more time: the curvature at the black hole horizon can be arbitrarily small. It depends on the mass of the black hole, the more massive the black hole, the smaller the curvature.

The curvature becomes strong towards the singularity. This is also where quantum gravitational effects are to be expected. Best,

Robert: You see the trash can symbol below your comments because you have logged in with your blogger ID, which allows you to delete your own comments. If you would post without logging in, you could not do that. As administrator of this blog, I have a trash can option below all comments. Best,

Thanks again for your patient explanation and I do remember the discussion. I can appreciate there are really two things that some consider with black hole models and information, with one being if they exists at all and the other if things truly can come to a singularity, with them not necessarily needing to be the same question or leading to the same conclusions.

Moving on to the solutions I must admit to still having problems. For example with all the conservative solutions you previously pointed out we have most of the information either released or permanently maintained (in a remnant or baby universe) at or near the end of the evaporation phase; so my confusion is as follows.

This whole entropy/information thing began with Bekenstein first demonstrating that the event horizon was not simply a boundary whose size is consistent with the mass/energy content of a black hole, yet also represents being a measure of the entropy there contained. What all the conservative models share in common being near the final phase of a black hole we end up with something having a low mass/energy plus low (thermal) entropy and yet with a extremely high information content.

So with such considerations we appear to have totally separated mass/energy along with entropy from the concept of information content. This truly is a strange state of affairs, for in what is all the information contained and even more importantly what if it is eventually released will it be restored to and how?

With the baby universe it seems to be a place having information with nothing for it to inform and with the remnant it seems similarly to have information being manifested as a physical entity of its own. In all the cases information conceptually seems to have exceeded from being considered the state of ordering of a system, as if to have us to imagine spin, orientation or what have you can exist independently of something to which it is to be applied. I simply have to ask myself. what exactly is it we are talking about here in the context of such consideration?

BTW At the moment, when your diagrams are based on relativity and radiative time arrows, it's virtually impossible to illustrate quantum gravity phenomena correctly by using of them - and vice-versa. I even needn't to look at it for knowing this.

Zephir: What I explained in the post is that an eternal black hole, as shown in the first diagram, can never be created at any finite time. It is the black hole shown in the 3rd diagram that is created from collapse of matter and can exist in reality. Best,

As long as qg effects do not result in non-local effects in areas with curvature smaller than the Planckian limit, you can use the diagrams to show the causal connections to the qg region, though not within it. If you assume spacetime is asymptotically flat and thus classical, this can be very useful. I'll come back to this in the next post. Best,

/*...eternal black hole.. can never be created at any finite time ....whereas.. black hole .. created from collapse of matter ... can exist in reality */

At the moment, when both black holes are containing singularity, they're indistinguishable each of other, because for creation of singularity you're required to have infinite amount of time (as you correctly exposed for yourself, but didn't understood).

Whereas without singularity we are discussing some dense stars or strangelets - but not black holes in common sense. We should categorize these things for further discussion to avoid neverending confusions. Is black hole without event horizon or central singularity (or both) still black hole in common sense?

A black hole is defined by the presence of the horizon, not the singularity. I would argue it doesn't have to be an event horizon. The statement that it takes an "infinite amount of time" to "create the singularity" is meaningless in general relativity without specifying what coordinates you are talking about, and therein lies your problem in understanding. Best,

/*..as long as qg effects do not result in non-local effects in areas with curvature smaller than the Planckian limit...*/Whole black hole is such nonlocal effect from exsintric perspective. You're just mixing different (dual) observational perspectives. From insintric perspective some curvature near black hole appears way, way larger, then from perspective of outer observer.

/*..the statement that it takes an "infinite amount of time" to "create the singularity" is meaningless in general relativity without specifying what coordinates you are talking about..*/

So why didn't specify it in your post [1:49 PM, November 28, 2009]? In fact it's always impossible from any coordinate system. If something is infinite, it would be always infinite from formal theory perspective, like the general relativity. You cannot have "smaller" or "larger" infinity.

Zephir: You have some very basic misunderstandings about general relativity. If you are really interested in the topic I strongly recommend you read a basic textbook. The spacetimes depicted above are geodesically incomplete. This means in the proper time of the geodesic it hits the singularity at a finite time. Yes, in the (unfortunately) commonly used Schwarzschild coordinates this time is infinite. Your statement

"something is infinite, it would be always infinite from formal theory perspective"

is simply wrong. An INVARIANT that is infinite is always infinite. So better make sure you talk about invariants before you make any claims. Best,

In the last diagram of an evaporating black hole, as the black hole evaporates and shrinks, at the very last moment will go through the NAKED singularity and disappear, presumably in explosion. Is not that weird?

Also, according to the diagram, the signal from a guy crossing the horizon (at any earlier time) will get released at that (very last) moment and reach the asymptotic observer. Is not that also weird? At the moment of the explosion of the black hole, I should be seeing signals of all the stuff that ever entered the balck hole!

1. Strominger: That was the problem we had to solve. In order to count microstates, you need a microscopic theory. Boltzmann had one–the theory of molecules. We needed a microscopic theory for black holes that had to have three characteristics: One, it had to include quantum mechanics. Two, it obviously had to include gravity, because black holes are the quintessential gravitational objects. And three, it had to be a theory in which we would be able to do the hard computations of strong interactions. I say strong interactions because the forces inside a black hole are large, and whenever you have a system in which forces are large it becomes hard to do a calculation.

The old version of string theory, pre-1995, had these first two features. It includes quantum mechanics and gravity, but the kinds of things we could calculate were pretty limited. All of a sudden in 1995, we learned how to calculate things when the interactions are strong. Suddenly we understood a lot about the theory. And so figuring out how to compute the entropy of black holes became a really obvious challenge. I, for one, felt it was incumbent upon the theory to give us a solution to the problem of computing the entropy, or it wasn't the right theory. Of course we were all gratified that it did.

3. TWO UNIVERSES of different dimension and obeying disparate physical laws are rendered completely equivalent by the holographic principle. Theorists have demonstrated this principle mathematically for a specific type of five-dimensional spacetime ("anti–de Sitter") and its four-dimensional boundary. In effect, the 5-D universe is recorded like a hologram on the 4-D surface at its periphery. Superstring theory rules in the 5-D spacetime, but a so-called conformal field theory of point particles operates on the 4-D hologram. A black hole in the 5-D spacetime is equivalent to hot radiation on the hologram--for example, the hole and the radiation have the same entropy even though the physical origin of the entropy is completely different for each case. Although these two descriptions of the universe seem utterly unalike, no experiment could distinguish between them, even in principle.

4.Consider any physical system, made of anything at all- let us call it, The Thing. We require only that The Thing can be enclosed within a finite boundary, which we shall call the Screen(Figure39). We would like to know as much as possible about The Thing. But we cannot touch it directly-we are restrictied to making measurements of it on The Screen. We may send any kind of radiation we like through The Screen, and record what ever changes result The Screen. The Bekenstein bound says that there is a general limit to how many yes/no questions we can answer about The Thing by making observations through The Screen that surrounds it. The number must be less then one quarter the area of The Screen, in Planck units. What if we ask more questions? The principle tells us that either of two things must happen. Either the area of the screen will increase, as a result of doing an experiment that ask questions beyond the limit; or the experiments we do that go beyond the limit will erase or invalidate, the answers to some of the previous questions. At no time can we know more about The thing than the limit, imposed by the area of the Screen.

Great post! One of my professors was discussing about this in one of his classes. I am a college sophomore with a dual major in Physics and Mathematics @ University of Canterbury in Christchurch, New Zealand. By the way, i came across these excellent physics flashcards. Its also a great initiative by the FunnelBrain team. Amazing!!!

The singularity is never outside the horizon, I don't know why you are talking about a naked singularity. Also, as I explicitly pointed out, the diagram is the standard one usually drawn, not the one I think makes the most sense. Please note that it does describe the last stage of the evaporation even though this is where effects of quantum gravity become strong and one thus can no longer apply the semi-classical treatment. More about that in a later post.

The signal from the guy crossing the horizon will end in the singularity. It cannot be "released" without violating locality. That is exactly the problem with the information loss. As I said, you can basically read it off the diagram now, which is why the diagrams are so useful. Best,

"The singularity is never outside the horizon, I don't know why you are talking about a naked singularity."

More correctly, when the mass of the blackhole has shrunk to a few Planck masses, the curvature at the horizon is large, quantum gravity effects are important, and we should not trust that part of the diagram. (At earlier times, the curvature at the horizon is small, and the Penrose diagram is trustworthy there.)

Instead, we should excise, from the diagram, all of those regions where the curvature is large (classically). That means, both the neighbourhood of the singularity, and the neighbourhood of the horizon at the final moments of the evaporation process.

"Also, as I explicitly pointed out, the diagram is the standard one usually drawn, not the one I think makes the most sense."

Once one excises (as one should), the regions of large curvature (where the classical description breaks down, and quantum gravity effects are important), the two are conformally equivalent (ie, the same).

I see some mention of the "other universe/s" that GR implies (due to disconnected nature of some ST diagrams, compare the lost region of Rindler coordinates)? I gather here, that current thinking is they don't really have to exist. They aren't related to QM multiworlds or other contexts for "other universes", but I heard of it reading about black hole event horizons etc. I can't imagine those other writers (including the type who should know) had no idea, so what was up with that?

Before a time classified as a Planck time, 10-43 seconds, all of the four fundamental forces are presumed to have been unified into one force. All matter, energy, space and time are presumed to have exploded outward from the original singularity. Nothing is known of this period.See:Before 1 Planck Time

As it still all seems illusive to me, it is certainly most enjoyable to see these points again as they are rehashed for better examination.

Thanks Bee.

I am trying to focus here so I found where I had put these points together for closer examination.

What are these notorious QG effects? Presumably semiclassical approximation breaks down but I want to see the exact calculation. I want to see the calculation at the turning point where during the last stages of the evaporation of the BH the strong gravitational background at the Horizon gives rise to a large vacuum polarization which in turn back reacts on the classical background producing an even stronger distorted gravitational background and thus a larger vacuum polarization etc... leading to inconsistencies and infinities. I would like to check such calculation specifically for a BH. It will be very instructive I think.

Blog:Once one excises (as one should), the regions of large curvature (where the classical description breaks down, and quantum gravity effects are important), the two are conformally equivalent (ie, the same).

Still, given how fragile entanglement is, Lloyd did not expect quantum illumination to ever work. But “I was desperate,” he recalls, keen on winning funding from a Defense Advanced Research Projects Agency’s sensor program for imaging in noisy environments. Surprisingly, when Lloyd calculated how well quantum illumination might perform, it apparently not only worked, but “to gain the full enhancement of quantum illumination, all entanglement must be destroyed,” he explains.

Lloyd admits this finding is baffling—and not just to him. Prem Kumar, a quantum physicist at Northwestern University, was skeptical of any benefits from quantum illumination until he saw Lloyd’s math. “Everyone’s trying to get their heads around this. It’s posing more questions than answers,” Kumar states. “If entanglement does not survive, but you can seem to accrue benefits from it, it may now be up to theorists to see if entanglement is playing a role in these advantages or if there is some other factor involved.”

As a possible explanation, Lloyd suggests that although entanglement between the photons might technically be completely lost, some hint of it may remain intact after a measurement. “You can think of photons as a mixture of states. While most of these states are no longer entangled, one or a few remain entangled, and it is this little bit in the mixture that is responsible for this effect,” he remarks.See:Quantum Entanglement Benefits Exist after Links Are Broken

Giotis: The point is that without knowledge of the full theory of quantum gravity one can't do the calculation. What one can do however is to determine the areas in which the effects should become important. That should be the case when the curvature at the horizon comes into the Planckian regime, which happens when the mass of the black hole is about the Planck mass (i.e. about 10^-5 gram if I recall correctly). What you can further do in absence of the full theory is to classify the possibilities. This is what was done in this paper, and now that I've explained what a causal diagram is, I can say somewhat more about the diagrams next time. Best,

Bee this is not what I meant. If you have BIRRELL & DAVIES check page 270. I mean something like the calculation described in the penultimate paragraph. This calculation does not need a QG theory. It is performed within the context of the semiclassical approximation.

I agree that the observer that is crossing the horizon ends up at the singularity, but the signal sent exactly when he crossed the singularity crosses the world line of an asymptotic observer at the same time as the signal of the explosion of the black hole (these two lines are actually on top of each other).

About the singularity, it is always enclosed by the horizon , except at the very last moment when the horizon disappears, at which moment becomes exposed.

'While topology has succeeded fairly well in mastering continuity, we do not yet understand the inner meaning of the restriction to differentiable manifolds. Perhaps one day physics will be able to discard it. '

We are now ready to do that. In fact it has been done and the results are so amazing that most theoretical physicists cannot even recognize or understand the successful completion of Einstein's 3-part relativity project: [1] Special Relativity (relativity of S-T for inertial frames); [2] General Relativity (relativity for inertial + accelerated frames); [3] discrete conformal relativity (discrete relativity of scale).

Or in Physics-speak, the Galaxy or the Black Hole at the (Spiral) Galaxy's center?

Click here to see was writer Claire moskowitz at space.com reports: the Black Hole.

I wish I could dig into this stuff more, but my current level of ignorance is such that ...

a) Black Hole theory is where quantum mechanics and general relativity BOTH break downandb) String theorists claim to have the answers, but then they always do, andc) Nobody knows for sure what the heck a REAL black hole is like (for the moment all analysis is almost purely mathematical), andd) The closest black hole if I remember correctly is 350 light years away, which is fine with me. :-)

Yeah, I'd have thought the black hole came first. Then the gas started swirling and plunging into the hole like water going down a plughole! Which lead to the formation of the galaxy structure. I mean, it seems pretty obvious you're not going to get something like a spiral galaxy without something pretty enormous in the middle around which everything spirals.

So note, Steven that one's level of ignorance should never stop one giving an opinion on subjects about which one knows very little! It never stops me!

I'd just like to point out that the coordinate "r" is *not* a measure of radial distance, not even outside the black hole. What it measures is the *area* of the sphere at a given point in the diagram. The advantage of thinking in this way is that you can still make sense of r even inside the event horizon where it is actually a time coordinate and not a spatial coordinate at all.

Now consider the point in Bee's diagram where the surface of the star meets the singularity. The singularity is the moment in time r = 0, hence at this point the surface area of the star has shrunk to zero. *But* it does not [in any simple way] follow that the *volume* of the star has also shrunk to zero! Because in these geometries, there is no simple relation between volume and area [cf the work of Steve Hsu]. So it is conceivable that the interior of the star does *not* age into the singularity, as everything inside the event horizon but outside the star must.

I'm not saying I believe that this happens, only that I don't know a proof that it can't. In the absence of such a proof, I would advise anyone standing on the surface of a collapsing star to burrow into the interior --- maybe you can survive that way. :-)

I think it would be fair to say that no one knows what r=0 means when it relates to time or space as we usually consider it, other than perhaps if one compares it to the perspective of a photon to say in terms of SR such conditions could be considered as its natural environment from the standpoint of any sub c entity (observer). In fact it could be used as an argument as why other dimensions beyond four or below one must have to be, for where else can it exist. Personally as I said before I prefer to look at this from the Cartesian view where zero being just a place (location) in mathematical space which separates positive from negative such as being a threshold rather than being a limit as its considered with calculus.

How about if we consider this passing through zero from a reductionist’s perspective, beginning with having only a four dimensional universe of which one being time. So to begin, if a sphere being the greatest amount of volume for the least amount of surface, then a negative sphere would be the least amount of volume for the greatest amount of surface, this being a circle. Then following down, a circle being the greatest amount of area for the least perimeter, would in the negative be the least amount of area for the greatest perimeter, which being a line so constructed as to guarantee it not able to return onto itself, of which the simplest being a straight one. Now if space is discrete in terms of units of measure and dimension (postulate), the equivalent in diminishing further would be to have the longest line having the greatest amount of discreteness being infinite or alternately all that’s allowed for in our universe, so the negative of this would be a line that has the least amount of discreteness as allowed for. The question is what this then represents as being, a dimensionless point or rather simply time?

As a follow up, let us consider our univerese as having higher dimensions as to ask why it so unusal to have it be possible, since as Einstein reminded distance is what you measure with rulers, while time is something one measures with clocks. So when considering more dimensions how can we say they don’t exist when we are surely uncertain as to what type they are, so to know how they might be measured?

Put another way from my perspective in terms of dimension when we rotate time one has a line. Rotate a line one has a circle. Rotate a circle one has a sphere. The question then being is time the primal dimension or is it also simply the result of rotating something more primal?

I just meant that Hsu's work is an example of the fact that there is no simple relation between area and volume --- as you said in your article about monsters, "In flat space-time the relation between the volume of an area and its surface is trivial, it's just Euclidean geometry. But not so if space-time is strongly curved!"

So the fact that a the surface area of the star [in the case you are discussing here] tends to zero does not imply that its volume tends to zero. The interior of the star might pinch off and become disconnected from the original spacetime. As I said, I'm not saying I believe this, only that it isn't obvious that this can't happen.

Note that you have drawn the surface of the star intersecting the lower of the two vertical lines labelled "r = 0". But the surface of the star might hit the *horizontal* line [the singularity] before that happens. That is the way it is drawn in Wald's textbook. Sorry I don't have Wald here with me so I can't tell you the page.

Yes, I agree with you regarding the surface/radius relation. I think we extensively commented on that in our paper too. Either way, in my diagram the surface doesn't hit r=0 before it hits the singularity, they all meet in the same point. I know that sometimes the surface is pictured as hitting the singularity on the horizontal line. I'm not sure this is uniquely specified without R(t), or is it? Either way, it doesn't make any difference for the formation/ evaporation/ information loss discussion. Best,

Bee, I have a simple question: Are quasars real in our time, or an evolutionary development in the history of our Universe?

My thinking (possibly wrong ... which is why I'm asking) is that since the closest quasar (active galaxy powered by a super-massive black hole) is on the order of billions of LIGHT YEARS away and no closer (probably a good thing), that they may have since evaporated and no longer exist anywhere at all right now, their "legacy" being the creation of galaxies and thereby: us.

The other possibility is that we're just fortunate to live in an area of space free of them. I don't know.

The evaporation for even a stellar mass black hole in accordance with Hawkings’ proposal is many times the current age of the universe so a supermassive one is much longer than that. What Hawkings’ referred to as primordial black holes having masses about that of a mountain have been calculated to have a lifetime of about 13 billion years and were speculated to perhaps become visible in the gamma ray burst which may be the mark of their end. There have been no such observations made so far which if this continues to be the case has implications for some early models of the universe or perhaps Hawking radiation or even more remotely black holes themselves. According to the current understanding supermmassive black holes will still exist in a universe that to the observer would be practically appear as empty and dark.

Just as a follow up is to say a more plausible explanation as to why there are no quasar like galaxies close at hand is that its consistent with the evolution of such structure. For instance the supermassive black hole that is suppose to exist at the center of our galaxy has long since swept up much of the readily available surrounding matter and thus is less active in terms of radiation production. Howver the more imminent fate of our galaxy has it colliding with Andromeda in a few billion years at which time each of their respective black holes may be given more stellar matter and gas to feed on. It would be an interesting time for experiment in terms of existing theory which I’m afraid all of us will be missing , that’s the price we pay for being Goldilocks creatures:-)

Funny thing, for the first time since I started re-learning all this Math and Physics stuff 14-1/2 months ago, I FINALLY met my first real Scientist in flesh and blood rather than on the internet last night, asked him my question, and he responded in the negative.

Dr. Shara told me that yes, massive quasar-size black holes do exist, in fact there is one very close by, by astronomical standards, that hasn't the fuel (yet) to shine like a quasar (which is a good thing for us), on the order of half a billion light years away.

He also stated, and this is important, that in spite of what Science News reports, black holes being the source of galaxies vs. vice-versa is STILL a very controversial issue in Astrophysics (that's "Astrophysics" NOT "Cosmology, or "Grand Astrophysical Speculation" as I call it, and I don't think I'm the only one).

I met Dr. Shara at my first Scientific "conference", really a roundtable discussion on the subject of "Time" at The Philoctetes Center in NYC on 82nd St. between 2nd and 3rd Aves. in NYC.

I think you would love the place, Phil. They're about tackling current speculative issues using an interdisciplinary approach with an emphasis on creativity and open discussion. The roundtable group consisted of, besides Dr. Shara, SciAm's George Musser (whom I also met for the first time ... he's a 44yr old EE just like Andrew Thomas, PhD EE), a Paleontologist, an Artist, and 3 others from different fields. I'll have more to say at my blog later today.

It should be noted that Philoctetes is a bit "controversial" as it's partially supported by the Templeton Foundation, a controversial organization it its own right thanks to the introduction of Theology. Woit and his followers savage both. I've had my fill of Theology for one lifetime, thank you, but I was there to see what the Physics guys had to say, and they had plenty.

Everything is on the table as far as I'm concerned regarding the big questions, so it's all good.

The session you mention sounds intriguing and yes it’s certainly something I wished I could have attended. Perimeter Institute in the past offered a similar thing where the researchers there would approach topics in general or at times even what they were currently working on. You might find it interesting they called them “Black Hole Sessions”. Unfortunately they curtailed these a few years ago which I personally think was a mistake in terms of outreach and building a bond between them and the public at large. I think the main problem with them at the time is that they didn’t have them challenge the attendees enough by means of introducing some structure by which those in attendance might actually learn things for themselves.

"I know that sometimes the surface is pictured as hitting the singularity on the horizontal line. I'm not sure this is uniquely specified without R(t), or is it?"

Well, the surface of the star can be represented by a particle falling freely. Where that particle hits the singularity depends on the initial conditions. [Think of particles falling into an ordinary black hole: they all hit the singularity at different places, depending on initial conditions.] Generically the surface will hit the singularity before reaching the vertical r = 0 line, as in Wald's diagram [page 156]. Wald shows the singularity continuing inside the star, and that is a sensible guess, but my point is that [as far as I know] this is just a good guess.

" Either way, it doesn't make any difference for the formation/ evaporation/ information loss discussion."

Well, if the interior of the star really could pinch off and make its own little universe, that *might* be relevant to information loss. But I'll stop with this because I don't really believe in this pinching off business anyway!

Either way, it doesn't make any difference for the formation/ evaporation/ information loss discussion.

Just to make comment on your most recent paper you coauthored with some Perimeter researchers, being I found it to be quite interesting. However I find it most interesting, not so much as it denies the existence of both singularities and an actual event horizon, rather it carries as a consequence that while there being no end for a black hole in regards to time with its end as to its demise , this still not having it preclude to it able in having a beginning. I guess I like it since with such a model it is consistence with what Cantor discovered, that being although an infinity is indeed endless, this does not have every infinity as being equal as to its origin or magnitude.

Also to make the observation that as of late you seem to be more prolific with your papers and to wonder what might be the cause as being resultant of more ideas of late, more time to express them,some consequence of your new enviroment or you in having reached some kind of flash point in regards to what has been on your mind for some time.

Thank you for your interest. It is a coincidence that my last two papers were finished almost at the same time. They both came about rather short notice, though the general topic in both cases is certainly something that's been on my mind for a longer time. Best,

Hi Bee. Thanks for maintaining a blog where an ex-CM experimentalist with some educational gaps can learn something about modern GR. (The jargon-dropping at Reference Frame really put me off, by the way).

I'm hundred persent sure I didn't see all the fine details of the causal diagram of radiating BH, but I'd like to understand few details:

1. if the BH is first growing, and then radiating away it's mass; is it so that in two different (horizontal lines on the diagram) hypersurfaces, the solution of EH is equal? Meaning that the area of EH finally gets smaller.Is this shown in the diagram? At least I cannot find it ;(

2. I have an intuitive feeling that when matter drops towards EH, the binding energy between the BH and matter is converted to kinetic energy (assuming no collisions etc); and as the total energy in any complete space-like hypersurface (I guess) needs to be equal, the rest mass of dropping matter thus need to be decreasing. What if the amount of rest mass, at the moment of passing through EH, were zero => all the rest mass is converted to kinetic energy, and the speed of matter is c. If so, then the dropping matter at the EH, should have light-like angle. And once inside the EH, the matter would still run at c.

1) There's no radial distances drawn in the diagram. The horizon remains timelike, but that doesn't mean the radial coordinate at the horizon remains constant when the black hole's mass is shrinking.

2) The rest mass of a particle is a conserved quantity, it never decreases. In GR, there's no such thing as a gravitational potential, there's only a curved background and particle's moving on geodesics in that background. Moreover, unlike the restmass, energies are observer-dependent quantities. When you think (for simplicity) of a photon approaching the horizon it becomes for the observer at asymptotic infinity more and more redshifted while for an also infalling observer no such effect is seen.